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0 sats \ 2 replies \ @beyond_turbulence OP 7 Dec \ on: The Bitcoin Topological Manifold: A Formalization of the Whitepaper's Geometry bitcoin
Theorem 3.1 (Proof-of-Work as Curvature).
The proof-of-work requirement for a block (finding a nonce such that
SHA256(block_header) < target) is equivalent to requiring the attached 2-cell to have scalar curvature R satisfying ∫_{f_i} R dA ≥ κ, where κ is the difficulty parameter.
The hash function acts as a random oracle mapping the block data to a point on a cryptographic manifold. The target defines a submanifold of acceptable solutions. Finding a nonce is finding a geodesic to this submanifold.
The expected work is exponential in the number of leading zero bits, analogous to the exponential decay of curvature in hyperbolic space.
Lemma 3.3 (Double-Spend as Boundary Dispute).
A double-spend attempt corresponds to two different 1-chains γ₁, γ₂ sharing a boundary vertex (the UTXO being spent) but with different coboundaries.
Only one can be included in the boundary of a valid 2-cell (block).
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Corollary 4.1 (Exponential Security). The probability that an attacker with less than 50% hash power can reverse a transaction with z confirmations drops exponentially in z, as stated in Section 11 of the whitepaper.
This follows from the Gambler's Ruin analysis in the whitepaper, interpreted as random walk on the 1-skeleton of the blockchain complex.
The deficit z is the distance between the honest chain tip and the attacker's fork tip on this graph.
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